https://file.aiquickdraw.com/mathbot/file/1726245690268fnf151rz.jpg

Let me help you simplify the expressions from the image. I'll go through each one by one:

### 1) \( 4 \sqrt{96} \)
To simplify:
\[
\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}
\]
Thus,
\[
4\sqrt{96} = 4 \times 4\sqrt{6} = 16\sqrt{6}
\]
So, the correct answer is **B) 16\(\sqrt{6}\)**.

### 2) \( 3 \sqrt{45} \)
To simplify:
\[
\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}
\]
Thus,
\[
3\sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5}
\]
So, the correct answer is **C) 9\(\sqrt{5}\)**.

### 3) \( \sqrt{27} \)
To simplify:
\[
\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}
\]

### 4) \( \sqrt{100} \)
\[
\sqrt{100} = 10
\]

### 5) \( 2\sqrt{6} - 2\sqrt{6} \)
This is straightforward:
\[
2\sqrt{6} - 2\sqrt{6} = 0
\]

### 6) \( 2\sqrt{3} + 2\sqrt{12} \)
To simplify \( \sqrt{12} \):
\[
\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}
\]
Thus:
\[
2\sqrt{3} + 2\sqrt{12} = 2\sqrt{3} + 2 \times 2\sqrt{3} = 2\sqrt{3} + 4\sqrt{3} = 6\sqrt{3}
\]

### 7) \( -3\sqrt{3} - 3\sqrt{12} + 3\sqrt{3} \)
Simplifying the middle term:
\[
\sqrt{12} = 2\sqrt{3}
\]
Thus:
\[
-3\sqrt{3} - 3 \times 2\sqrt{3} + 3\sqrt{3} = -3\sqrt{3} - 6\sqrt{3} + 3\sqrt{3}
\]
\[
= (-3 - 6 + 3)\sqrt{3} = -6\sqrt{3}
\]

### 8) \( 2\sqrt{24} + 2\sqrt{6} - 2\sqrt{6} \)
Simplifying \( \sqrt{24} \):
\[
\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}
\]
Thus:
\[
2\sqrt{24} + 2\sqrt{6} - 2\sqrt{6} = 2 \times 2\sqrt{6} + 2\sqrt{6} - 2\sqrt{6}
\]
\[
= 4\sqrt{6} + 2\sqrt{6} - 2\sqrt{6} = 4\sqrt{6}
\]

### 9) \( \sqrt{10} \cdot \sqrt{6} \)
Using the multiplication property of square roots:
\[
\sqrt{10} \cdot \sqrt{6} = \sqrt{10 \times 6} = \sqrt{60}
\]

### 10) \( \sqrt{10} \cdot 3\sqrt{6} \)
To simplify:
\[
\sqrt{10} \cdot 3\sqrt{6} = 3 \cdot \sqrt{10 \times 6} = 3\sqrt{60}
\]

### 11) \( \sqrt{6} \cdot \sqrt{5} \)
Using the multiplication property of square roots:
\[
\sqrt{6} \cdot \sqrt{5} = \sqrt{6 \times 5} = \sqrt{30}
\]

### 12) \( \sqrt{5} (\sqrt{2} + \sqrt{6}) \)
Distribute \( \sqrt{5} \):
\[
\sqrt{5} (\sqrt{2} + \sqrt{6}) = \sqrt{5} \cdot \sqrt{2} + \sqrt{5} \cdot \sqrt{6} = \sqrt{10} + \sqrt{30}
\]

### 13) \( \frac{\sqrt{16}}{\sqrt{25}} \)
Simplify both square roots:
\[
\frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5}
\]

### 14) \( \frac{\sqrt{16}}{5\sqrt{25}} \)
Simplify both square roots:
\[
\frac{\sqrt{16}}{5\sqrt{25}} = \frac{4}{5 \times 5} = \frac{4}{25}
\]

---

Would you like any further clarification on these steps or details on any specific problem? Here are some related questions you might want to explore:

1. How do you simplify square roots that involve multiplication?
2. What is the general property of square roots when it comes to division?
3. How do you distribute square roots in expressions like question 12?
4. Can all square roots be simplified, and if not, why?
5. What are some common mistakes to avoid when simplifying radical expressions?

**Tip**: When simplifying radicals, always check if the number under the square root has any perfect square factors (e.g., 4, 9, 16, etc.) to simplify the process.

Simplify various radical expressions involving square roots. Learn step-by-step methods using the multiplication property of radicals and distributive property. Suitable for high school students in Grades 9-10.

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